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Questions tagged [topology]

In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.

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Are there closed simply connected 2D manifolds that do not require a third dimension?

In the context of cosmology, space is commonly described as potentially having a global curvature that can be positive, zero, or negative. A common way that textbooks describe positive curvature is by ...
scottduhnam's user avatar
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Can part of space be causally disconnected from the rest of the universe by being surrounded by black holes? [duplicate]

Is it possible for black hole event horizons to overlap and form a spherical wall around an island of space (that's not inside a black hole) while still being causally disconnected from the rest of ...
user3624007's user avatar
2 votes
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Instantons in the Global $O(2)$ Model (Compact scalar field) - Polyakov textbook

This question is related with Polyakov, "Gauge Fields and Strings" section 4.2 In section 4.2, partition function is \begin{equation} Z=\sum_{n_{x,\delta}}\int_{-\pi}^{\pi}\prod_x\frac{d\...
zahra's user avatar
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Action principle dependent on spacetime-topology?

Consider the Lagrangian density $$L(\phi, \nabla \phi, g) = g^{\mu \nu} \nabla_{\mu} \phi \nabla_{\nu} \phi$$ If one varies the action as usual, then one finds the equation $$\delta S = \int_{\mathcal{...
Octavius's user avatar
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What is the importance of $SU(2)$ being the double cover of $SO(3)$?

To my understanding, it is important that $SU(2)$ is (isomorphic to) the universal cover of $SO(3)$. This is important because $SU(2)$ is then simply-connected and has a Lie algebra isomorphic to $\...
Silly Goose's user avatar
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1 vote
2 answers
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The topology of planets [duplicate]

Just a curiosity: Let $g \in \mathbb{Z}_{>0}$. Is it possible for a planet of topological genus $g$ to exist? For example, is there any contradiction (from the point of view of physics) in assuming ...
numberwat's user avatar
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What is the topology of a spacelike cross-section of a null hypersurface?

I'm studying the peeling-off behaviour of zero rest-mass fields, as described in Penrose's paper. In it, he talks about the boundary $\mathscr{I}$ of the conformal completion of an asymptotically ...
Smikkelma's user avatar
2 votes
3 answers
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What is the dual asymptotic spacetime of a CFT on a particular flat manifold?

According to AdS/CFT correspondence, the dual theory of a boundary CFT on flat spacetime is defined on an asymptotically AdS spacetime. The nature of the bulk spacetime depends on the topology of the ...
Sanjana's user avatar
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2 answers
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Are $i^\pm$ and $i^0$ codimension 1 surfaces?

Standard textbooks like Carroll's say that spatial and temporal infinities in Minkowski space Penrose diagram are points. But on the footnote in pg. 3 of some draft notes on Celestial holography by ...
Sanjana's user avatar
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Conformal compactification of AdS spacetime

In this paper https://homes.psd.uchicago.edu/~ejmartin/course/JournalClub/Basic_AdS-CFT_JournalClub.pdf, page 2, the authors state "The boundary of the conformal compactified $AdS_{d+1}$ is ...
Βασίλης Γερμανίδης's user avatar
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9 answers
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Why are spherical shapes so common in the universe?

I have a simple question. Why are most objects in the observable universe spherical in shape? Why not conical, cubical, cuboidal for instance? I am furnishing a few points to justify this statement: ...
Ishaan's user avatar
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2 answers
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How does one prove that the electric field is conservative when it is not defined on simply-connected regions?

I apologize for the ignorance and the rough English in advance, I have an issue understanding how to match both what happens in physics and what I am seeing in calculus. In my calculus class we ...
Some random guy's user avatar
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1 answer
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What's the relations between Quantum spin liquid and Quantum magnetism? [closed]

I am a fourth years undergraduate student. Recently, I am seeking that my research direction for my upcoming graduate program, and I found that my tutor is working that direction (as shown in the ...
Tierisches Gift's user avatar
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1 answer
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What objects are solutions to the Einstein Field Equations?

The usual way the solutions of the Einstien Field Equations are introduced is by saying they are (pseudo-) riemannian metrics that satiafy the diff equations for a given EM Tensor. My question is: ...
emilio grandinetti's user avatar
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Why the unit of quantum information for anyons systems should be the qubit?

I'm starting to learn more about anyons systems. I took a read on this article which is an introduction to topological quantum computing, and also took a look in other places like forums and some ...
Lucas Sievers's user avatar
2 votes
1 answer
78 views

Topological behavior (or asymptotics at infinity) of gauge fields assumed in Fujikawa method

Chiral anomaly is computed very elegantly by Fujikawa method, which is also presented in Section 22.2 of Weinberg QFT textbook volume 2 or wikipedia. Here, the underlying spacetime is assumed to be $\...
Keith's user avatar
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Exact definition of topological non-identical diagrams

It is often said that Feynman diagrams for fermions do not have symmetry factors. Consider I have a spinless fermionic quantum many-body system with Hamiltonian: $$H=\int_{r}\psi^{\dagger}(r)\frac{\...
John 's user avatar
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1 answer
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Visualization of a gauge field with non-null winding number

In QCD you may add the term $\mathcal{L}_{\theta} = \theta\dfrac{g^2}{16\pi^2} \text{Tr}F\tilde{F}$, which turns out to be a total derivative. Now, it can be proven that the action of this lagrangian ...
Gabriel Ybarra Marcaida's user avatar
2 votes
0 answers
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Understanding topological censorship: is something wrong with this example?

Earlier today I was discussing with someone about the possibility of space (not spacetime) being a torus, and how this is different from a sphere. For simplicity, let us assume spacetime is of the ...
Níckolas Alves's user avatar
2 votes
1 answer
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Abelian Chern-Simons large gauge transform

My question concerns the $U(1)$ Chern-Simons theory with the action $$S = \frac{k}{2\pi}\int A\wedge \mathrm{d}A.$$ In my lecture, it is stated that: A large gauge transformation involves taking $A\...
shamwowexcitante's user avatar
4 votes
0 answers
167 views

Anyons and Elementary particles in 2D [closed]

I'm doing my master's degree and I'm starting to learn more about Anyons. I want to understand more deeply why they can exist and how. I've done some research on the internet and found this question ...
Lucas Sievers's user avatar
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0 answers
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How to verify the compatibility condition for Berry's connection?

I am reading Mikio Nakahara's Geometry, Topology and Physics. In Chapter 10, he defines the Berry's connection one-form on the $U(1)$ bundle as $$ \mathcal{A}=\left\langle\mathbf{R}\right|d\left|\...
Zhicheng ZHANG's user avatar
2 votes
2 answers
116 views

Solutions to Maxwell's equations with $dF=0$ but $F \neq dA$ -- can the new solutions be summarized by considering only the vacuum equations?

I am trying to learn a bit about differential forms. I saw a question and answer noting that the homogeneous Maxwell equations can be written as $dF=0$. However, as noted there, depending on the ...
user196574's user avatar
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1 vote
0 answers
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How to numerically calculate Zak phase for SSH3 model?

The k-space hamiltonian of SSH3 model with nearest neighbour hopping is given by H(k)= \begin{bmatrix} 0 & u & w e^{-ika} \\ u & 0 & v \\ w e^{ika} & v & 0 \end{bmatrix} ...
SUMANTA SANTRA's user avatar
1 vote
0 answers
33 views

Is the Berry phase at a degeneracy trivial or non-trivial?

This question is about the original paper on Berry phases by M. Berry (1984). There, the Berry phase $\phi_{B}$ is defined as $$ \phi_{B}= \oint_\mathcal{C}\underbrace{ \left\langle n(\mathbf{R}) \...
xabdax's user avatar
  • 249
10 votes
8 answers
6k views

Is there a true one-dimensional object? [closed]

I'm reviewing and expanding my knowledge of dimensions. We live in three spatial dimensions but, apart from volume, we also have the concept of surface and curve. However, if you write a line on paper,...
jmazaredo's user avatar
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1 vote
0 answers
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2-dimensional connected Lorentz group [closed]

Consider the connected Lorentz group $SO(1,1)^{\uparrow}$. I was wondering if someone could help me about showing that $SO(1,1)^{\uparrow}\cong \mathbb{R}\times \mathbb{Z}_2$. I just need a hint.
Mahtab's user avatar
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2 answers
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What is the manifold topology of a spinning Cosmic String?

Given the following metric which is that of a rotating Cosmic String: $$g=-c^2 dt^2 + d\rho^2 + (\kappa^2 \rho^2 - a^2) d\phi^2 - 2ac d \phi dt + dz^2.$$ can one determine the manifold topology ...
Bastam Tajik's user avatar
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0 votes
1 answer
149 views

Black holes, singularities and topology in relativity

General relativity is defined on a base manifold which, viewed as a topological space, is simply connected (which means there's no holes). However, we know that inside a black hole there's a ...
Tomás's user avatar
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2 votes
1 answer
63 views

Original source "GF92" for two drawings of Dirac belt trick

I am trying to track down the original source/artist of these two drawings of the Dirac belt trick (see link below) to use in my thesis (which is in mathematics, but I believe these pictures likely ...
1 vote
0 answers
28 views

Tenfold way symmetry classification for systems with pseudomomentum

For classifying Hamiltonians $H(\vec{k})$ of topological insulators/superconductors in the tenfold way, one has to see whether the Hamiltonians obeys (disobeys) symmetries of the following type (let's ...
Dave Force's user avatar
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0 answers
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How does time-reversal symmetry breaking work in a Weyl crystal?

In a Weyl crystal, we can break the TRS (Time-Reversal Symmetry) by introducing FM or AFM atoms, and this breaking occurs due to the emergence of spontaneous magnetization in the material (I think), ...
Gabriel Elyas's user avatar
1 vote
1 answer
50 views

Preferred state of motion from topology of certain spacetimes

A (1+1)-dimensional spacetime where the spatial dimension wraps around on itself (so it has the topology of $E^1\times S^1$) has a preferred state of motion, even though it is everywhere locally ...
Matt Dickau's user avatar
0 votes
1 answer
52 views

Is there a topological invariance/winding number for non-translation invariance system?

My question is: Is there a topological invariance/winding number for non-translation invariance system? For example, if we modify the interacting parameter in SSH model, such that it depends on the ...
feng lin's user avatar
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1 vote
0 answers
306 views

The Lebesgue covering dimension of the Cosmic String interval topology

Take the spacetime $(M,g)$ that satisfies Einstein's Field Equations exactly where $g$ is locally: $$g= - c^2 dt^2 + d \rho^2 + (\kappa^2 \rho^2 - a^2) d \phi^2 - 2 ac d\phi dt + dz^2 \ $$ in the ...
Bastam Tajik's user avatar
  • 1,212
2 votes
0 answers
57 views

How can I construct a projective representation when the group is not simply connected?

S. Weinberg, in his book "The quantum theory of fields", states this theorem (page 83): The phase of any projective representation $U(T)$ of a given group can be chosen so that $\phi =0$ if ...
Mahtab's user avatar
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1 vote
1 answer
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Why future infinity have no future end points?

I am studying Hawking's area theorem from the book the large scale structure of spacetime by Hawking and Ellis. At the end of page#318, it said: null geodesic generators of future infinity have no ...
Talha Ahmed's user avatar
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1 answer
151 views

How does null infinity differ from ordinary infinity?

Null infinity is the diagonal lines on the edge of a Penrose diagram. It seems to be the place where beams of light go if they never bump into anything, but only light can go there. It appears to be ...
Miss Understands's user avatar
2 votes
1 answer
123 views

Could the universe have a form of a $T^3$-torus?

Cosmological measurements suggest that we live in a flat universe. However, what might be less clear is its topology. So could the flat universe have the form of a $T^3$-torus, i.e. the torus whose ...
Frederic Thomas's user avatar
2 votes
0 answers
135 views

Einstein's gravity Lagrangian invariance under the change of differential structure

I came across an article claiming the appearance of singularities in the energy-momentum tensor $T_{\mu \nu}$ as a result of changing the differential structure: I wonder what symmetry or current (in ...
Bastam Tajik's user avatar
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-4 votes
2 answers
138 views

What model tells us there is nothing outside the universe?

Is there an existing model or theory that shows there is nothing outside of the universe that interacts with anything inside the universe? Or to put it in other words, is there a model or theory ...
foolishmuse's user avatar
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3 votes
1 answer
161 views

Divergence of the Berry connection

Given the Berry connection \begin{equation} \boldsymbol{\mathcal{A}}(\mathbf{R}) = i \langle u(\mathbf{R}) | \nabla_\mathbf{R} | u(\mathbf{R}) \rangle, \end{equation} the Berry curvature can be ...
Lucas Baldo's user avatar
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0 votes
0 answers
19 views

How is the coupling between order parameter and strain determined from symmetry?

In the article Ehrenfest Relations for Ultrasound Absorption in Sr2RuO4, Sigrist M., Progress of Theoretical Physics, Vol. 107, No. 5, May 2002 A superconductor with proposed p-wave pairing and ...
Nitzan R's user avatar
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0 votes
0 answers
37 views

Conformal symmetry and group in arbitrary dimensions [duplicate]

As far as i understand, the full symmetry of relativity is conformal symmetry. This is represented by the conformal group $ \operatorname{Conf}(1, 3) $ Of Minkowski spacetime which is $ \mathbb{R}^{1, ...
Tomás's user avatar
  • 309
3 votes
2 answers
616 views

Why radial quantization gives different spectrum?

For example we work with 1+1D massless free boson, in canonical quantization we allow creation operators at any momentum so the Hamiltonian has continuous spectrum. But if we conformally map to a ...
Peter Wu's user avatar
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1 vote
0 answers
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JT gravity metric - solution to the dilaton equations of motion

I am reading Closed universes in two dimensional gravity by Usatyuk1, Wang and Zhao. The question is not too technical, it is about the solutions to the equations of motion that result from the ...
schris38's user avatar
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1 vote
2 answers
127 views

Understanding Exceptional Points

Exceptional points occur generically in eigenvalue problems that depend on a parameter. By variation of such parameter (usually into the complex plane) one can generically find points where ...
ZHENGYAO HUANG's user avatar
0 votes
1 answer
74 views

Are there reasonable models of Earth's surface as the space $\mathbb{R}P^2$? [closed]

Everyone knows that the Earth's surface is a 2-sphere, or a geoid. Flat earthers propose that Earth's surface is a disk or some variation of that, and there is lots of discussion on why its not true ...
Rumpelstiltskin's user avatar
2 votes
0 answers
59 views

Examples of spacetimes that are asymptotically flat at future timelike infinity

There are interesting non-trivial examples of spacetimes which are asymptotically flat at null and spacelike infinities. For example, the Kerr family of black holes satisfies these conditions. However,...
Níckolas Alves's user avatar
0 votes
0 answers
30 views

What is the Chern number of twisted bilayer graphene without hBN substrate?

I am approaching at the study of topological materials and I need help to understand the role of Berry curvature in the twisted bilayer graphene. The raise up of the Moire lattice and consequently the ...
Andrea Zacheo's user avatar

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